# Mahāvīra (mathematician)

**Mahāvīra** (or **Mahaviracharya**, "Mahavira the Teacher") was a 9th-century Jain mathematician born in the present day city of Gulbarga, Karnataka ^{[1]}, in southern India.^{[2]}^{[3]}^{[4]} He authored *vedh granth * (*Ganita Sara Sangraha*) or the Compendium on the gist of Mathematics in 850 CE.^{[5]} He was patronised by the Rashtrakuta king Amoghavarsha.^{[5]} He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.^{[6]} He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.^{[7]} He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle.^{[8]} Mahāvīra's eminence spread throughout South India and his books proved inspirational to other mathematicians in Southern India.^{[9]} It was translated into Telugu language by Pavuluri Mallana as *Saar Sangraha Ganitam*.^{[10]}

He discovered algebraic identities like *a*^{3} = *a* (*a* + *b*) (*a* − *b*) + *b*^{2} (*a* − *b*) + *b*^{3}.^{[4]} He also found out the formula for ^{n}C_{r} as

[*n* (*n* − 1) (*n* − 2) ... (*n* − *r* + 1)] / [*r* (*r* − 1) (*r* − 2) ... 2 * 1].^{[11]} He devised a formula which approximated the area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number.^{[12]} He asserted that the square root of a negative number does not exist.^{[13]}

## Rules for decomposing fractions[edit]

Mahāvīra's *Gaṇita-sāra-saṅgraha* gave systematic rules for expressing a fraction as the sum of unit fractions.^{[14]} This follows the use of unit fractions in Indian mathematics in the Vedic period, and the Śulba Sūtras' giving an approximation of √2 equivalent to .^{[14]}

In the *Gaṇita-sāra-saṅgraha* (GSS), the second section of the chapter on arithmetic is named *kalā-savarṇa-vyavahāra* (lit. "the operation of the reduction of fractions"). In this, the *bhāgajāti* section (verses 55–98) gives rules for the following:^{[14]}

- To express 1 as the sum of
*n*unit fractions (GSS*kalāsavarṇa*75, examples in 76):^{[14]}

rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /

dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //

When the result is one, the denominators of the quantities having one as numerators are [the numbers] beginning with one and multiplied by three, in order. The first and the last are multiplied by two and two-thirds [respectively].

- To express 1 as the sum of an odd number of unit fractions (GSS
*kalāsavarṇa*77):^{[14]}

- To express a unit fraction as the sum of
*n*other fractions with given numerators (GSS*kalāsavarṇa*78, examples in 79):

- To express any fraction as a sum of unit fractions (GSS
*kalāsavarṇa*80, examples in 81):^{[14]}

- Choose an integer
*i*such that is an integer*r*, then write - and repeat the process for the second term, recursively. (Note that if
*i*is always chosen to be the*smallest*such integer, this is identical to the greedy algorithm for Egyptian fractions.)

- To express a unit fraction as the sum of two other unit fractions (GSS
*kalāsavarṇa*85, example in 86):^{[14]}

- where is to be chosen such that is an integer (for which must be a multiple of ).

- To express a fraction as the sum of two other fractions with given numerators and (GSS
*kalāsavarṇa*87, example in 88):^{[14]}

- where is to be chosen such that divides

Some further rules were given in the *Gaṇita-kaumudi* of Nārāyaṇa in the 14th century.^{[14]}

## See also[edit]

## Notes[edit]

**^**http://firstip.org/legendary-scientists/mahaviracharya8th-century**^**Pingree 1970.**^**O'Connor & Robertson 2000.- ^
^{a}^{b}Tabak 2009, p. 42. - ^
^{a}^{b}Puttaswamy 2012, p. 231. **^**The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88**^**Algebra: p.43**^**Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122**^**Hayashi 2013.**^**Census of the Exact Sciences in Sanskrit by David Pingree: page 388**^**Tabak 2009, p. 43.**^**Krebs 2004, p. 132.**^**Selin 2008, p. 1268.- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}Kusuba 2004, pp. 497–516

## References[edit]

- Bibhutibhusan Datta and Avadhesh Narayan Singh (1962).
*History of Hindu Mathematics: A Source Book*. - Pingree, David (1970). "Mahāvīra".
*Dictionary of Scientific Biography*. New York: Charles Scribner's Sons. ISBN 978-0-684-10114-9.CS1 maint: ref=harv (link) - Selin, Helaine (2008),
*Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures*, Springer, ISBN 978-1-4020-4559-2 - Hayashi, Takao (2013), "Mahavira",
*Encyclopædia Britannica* - O'Connor, John J.; Robertson, Edmund F. (2000), "Mahavira",
*MacTutor History of Mathematics archive*, University of St Andrews. - Tabak, John (2009),
*Algebra: Sets, Symbols, and the Language of Thought*, Infobase Publishing, ISBN 978-0-8160-6875-3 - Krebs, Robert E. (2004),
*Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance*, Greenwood Publishing Group, ISBN 978-0-313-32433-8 - Puttaswamy, T.K (2012),
*Mathematical Achievements of Pre-modern Indian Mathematicians*, Newnes, ISBN 978-0-12-397938-4 - Kusuba, Takanori (2004), "Indian Rules for the Decomposition of Fractions", in Charles Burnett; Jan P. Hogendijk; Kim Plofker; et al. (eds.),
*Studies in the History of the Exact Sciences in Honour of David Pingree*, Brill, ISBN 9004132023, ISSN 0169-8729